Due to the centralized location of Big Eightmile Peak, there are numerous approaches to reach it. All of the routes have their own perks, but I think the trail following the Middle Fork of Little Timber Creek is probably the most appealing. It provides the broadest variety of terrain, passing three scenic lakes and a number of other central Lemhi Peaks. The route follows a trail most of the way, with a bit of Class 2 cross-country travel and a couple scramblish type steps on the northeast ridge of the peak.
From Leadore, turn west from the center of town on Lee Creek Road. After 0.8 miles, turn left on Timber Creek Road. This road is paved a first, but changes into to a gravel road after a couple miles. Once on Timber Creek Road, follow the road most traveled through a couple forks, following signs for Stone Reservoir and/or Timber Creek Reservoir (these two names seem to be used interchangeably for the lake near the trailhead). At just over 10 miles form Leadore, you'll see the reservoir. Continue on the road that rises above the reservoir to the south and continue to the end of the line where the road meets the Middle Fork of Little Timber Creek, just over 11 miles from Leadore.
A passenger car might be able to make it to the trailhead, but consider a rental car or borrow one from a relative that you don't like. High clearance is recommended, four wheel drive isn't necessary during the summer months.
Once at the trailhead, the trail crosses the creek and stays on the north side for just over 2.5 miles. No danger of losing you way here since its well worn due to being open to motorcycles and ATV's. (The trail is in very good shape in spite of this, and probably sees limited motorized travel because the remoteness of the area). When the main trail makes the stream crossing at 2.5 miles, you can continue on it "the long way" to Yellow Lake, or look for a faint path veering off to the right just before the crossing. This unmaintained trail winds its way to Middle Fork Lake (9157') which boasts some great campsites and trout fishing.
From Middle Fork Lake make your way northwest up and around to a finger of talus, then contour west to the lowest point on the ridge. Once on the ridge, you can make an easy side trip up to nearby "Slate Roof Peak" (10507'), or begin a gentle descent to Yellow Lake. Yellow Lake is about 4.5 miles from the trailhead if you take the shortcut via Middle Fork Lake.
The route from Yellow Lake follows a trail shown on the USGS map. The trail hasn't been maintained for several years and can be difficult to locate and follow. The best bet is to follow the Yellow Lake outlet a couple hundred yards then cross the stream and wander cross-country until you find a path. Once on it, there are occasional cairns and blazes. Follow this for a while with your next target being the 9892' saddle on the northeast ridge of Big Eightmile Peak. Just below the trail lies a secluded unmapped lake. The 9892' saddle is just over a mile from Yellow Lake.
Once to the saddle, the summit isn't quite visible. From here, leave the trail and head up the ridgeline. The easiest going is on the crest of the ridge. There's some minor scrambling before reaching a bit of a plateau, then make the final easy climb to the summit. From the top, there are great views of the surrounding Lemhi's, the Beaverheads to the east, and the Lost River Range to the southwest.
Its just over 6.5 miles from the trailhead to the summit of Big Eightmile Peak, which is about 2900' higher than the starting point.
No technical gear required, just dress for the weather.
Be prepared for possible afternoon thundershowers.