The word, "eureka" is a direct transliteration of a Biblical Greek word, "eurisko." "Eureka" is the 1st person singular in the Perfect tense and would literally translate, "I have found!" Perhaps this peak was named by an excited miner who found whatever treasure he was looking for.
From either of the two possible parking locations, walk on up the Hermit Pass road to the pass summit. That will be about the easiest 1.3 miles of your day. From the pass, walk up and over Hermit Peak. That will count as a ranked 13er summit. It will only take about 15 - 20 minutes from the pass to reach Hermit's summit about 330 feet uphill.
From Hermit Peak, proceed SSE descending the steep and rocky ridge to a saddle at 12,700 ft. where rock gives way to a broad tundra-covered meadow. Either contour around or go directly over the unranked Pt.13,005 to yet another tundra bench/meadow. Then begin the final approach to Eureka on an ever steepening slope where tundra will give way to almost all rocky rubble. Once on the main summit ridge, you'll find two summits of nearly equal height separated by a short stretch of rocky ridge. We hit both high points just to be sure, but to us, the western summit appeared to be the highest, even though Google Earth seems to indicate the easternmost is highest. It certainly won't hurt that much to tag both since there's only a little bit of scrambling in between.
From the summit of Eureka, you may want to scan the distant Unnamed 13,054 to the SW. If climbing all the 13ers is on your wish list, then that's a summit you'll be pursuing some day. There's some very remote country in those basins and valleys to the west. For a descent of Eureka and return to your vehicle, we dropped back down the north ridge of Eureka to the tundra meadow between Eureka and Pt. 13,005, then walked over to the eastern edge of the meadow and began a rugged descent down through broken cliffs and rocks to steep, tundra-covered and flower-laden slopes further down as we neared Eureka Lake. After you've stopped to admire the beautiful lake, follow a mostly tundra bench to the north and Horseshoe Lake, at one point having to cross a boulder field, where a trail that connects the two lakes may disappear for a while. From Horseshoe Lake, make the quick ascent back up to the road and your awaiting vehicle. This loop hike of Hermit Peak and Eureka Mtn., could be completed in about 2.5 to 4 hours.
Links to other information, routes & trip reports for this peak that may be helpful.