Bulgars were originally a Turkic-speaking tribe that then merged with Slavic tribes they conquered in the area of present-day Bulgaria and took over their language. The Turks then conquered them in the 14th C., influencing the language for several centuries so that modern Bulgarian has many words from Greek and Turkish.
Sevda is from the Turkish root sevgi/sevi meaning 'love' and would mean 'loved'. The 'precious' meaning you got inevitably relates to the expanded translation 'loved, cherished' as one would cherish and love something that was precious to them. It all leads back to the same, ultimately Turkish, source.
My Turkish ex always called me sevgilim (sounds much prettier in spoken Turk - sev-jhee-lim) which means 'darling' or 'beloved', from the same sevgi root. Clearly your daughter was very much loved by whoever named her in Bulgaria :o)